Properties

Label 4-1350e2-1.1-c3e2-0-15
Degree $4$
Conductor $1822500$
Sign $1$
Analytic cond. $6344.53$
Root an. cond. $8.92482$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 60·11-s + 16·16-s + 278·19-s − 468·29-s − 110·31-s + 276·41-s − 240·44-s + 157·49-s + 792·59-s + 46·61-s − 64·64-s + 408·71-s − 1.11e3·76-s + 1.41e3·79-s + 1.63e3·89-s − 2.55e3·101-s − 1.18e3·109-s + 1.87e3·116-s + 38·121-s + 440·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.64·11-s + 1/4·16-s + 3.35·19-s − 2.99·29-s − 0.637·31-s + 1.05·41-s − 0.822·44-s + 0.457·49-s + 1.74·59-s + 0.0965·61-s − 1/8·64-s + 0.681·71-s − 1.67·76-s + 2.01·79-s + 1.94·89-s − 2.51·101-s − 1.04·109-s + 1.49·116-s + 0.0285·121-s + 0.318·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1822500\)    =    \(2^{2} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(6344.53\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1822500,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.795004265\)
\(L(\frac12)\) \(\approx\) \(3.795004265\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p^{2} T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 157 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 30 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3238 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 8062 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 139 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 12530 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 234 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 55 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 64825 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 138 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 156205 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 73690 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 188854 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 396 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 23 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 397222 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 204 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 300553 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 709 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 62030 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 816 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1006321 T^{2} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.333771564473443982250173851948, −9.198332325488581750882798452470, −8.918711807810749363964605666697, −8.089315436676468617745924990323, −7.80517189333905903915727446340, −7.45412282331189108906789588007, −6.95858885433513371841813469362, −6.79520705165701299812025387110, −5.84408649574187243833065410715, −5.79654734428314153900682027796, −5.19142333856758056245891650680, −5.03118314197903679268832560075, −4.06120804337144436867189475944, −3.84046592406864886065674703122, −3.52749027710271547491704356211, −2.98233355757029309152521709874, −2.14823276783165838146648620826, −1.56476808537157535037200275645, −0.996711207653808070261584069649, −0.54816388286798262476736766119, 0.54816388286798262476736766119, 0.996711207653808070261584069649, 1.56476808537157535037200275645, 2.14823276783165838146648620826, 2.98233355757029309152521709874, 3.52749027710271547491704356211, 3.84046592406864886065674703122, 4.06120804337144436867189475944, 5.03118314197903679268832560075, 5.19142333856758056245891650680, 5.79654734428314153900682027796, 5.84408649574187243833065410715, 6.79520705165701299812025387110, 6.95858885433513371841813469362, 7.45412282331189108906789588007, 7.80517189333905903915727446340, 8.089315436676468617745924990323, 8.918711807810749363964605666697, 9.198332325488581750882798452470, 9.333771564473443982250173851948

Graph of the $Z$-function along the critical line